# Integrate the exponential of sum of circular differences?

Given positive integer $N$ and parameters $T>0$, $a$, $b$, what is

$\int_{t_1=0}^T \cdots \int_{t_N=0}^T e^{a(t_1+\cdots+t_N)+b(|t_1-t_2|+\cdots+|t_{i-1}-t_i|+|t_N-t_1|)} dt_1 \cdots dt_N$ ?

Any thought will be very much appreciated!

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Interesting looking integral. Could you say a bit more about where it comes from, what the motivation for the problem is, whether or not estimates/asymptotic formulas instead of an exact formula would be useful for you, and so on? – Yemon Choi Jun 28 '13 at 1:40
Yemon, thanks for your comment. I do need an exact formula. This integral comes from calculating high moments of a functional of some non-stationary Gaussian process. I don't have enough space to describe details of the model, and therefore I singled out the core math problem. – Isley Jun 28 '13 at 17:04
I don't believe you didn't do $N=1,2,3$. Can you show what you got there? Essentially, the cube splits into $N!$ simplices of equal volume (according to the order of variables) and the integral over each simplex is not a big headache. The question is if there are some interesting cancellations. – fedja Sep 3 '13 at 17:39